Cambridge IGCSE Mathematics - Additional 0606 — 2006 Oct/Nov Paper 1 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/01 Paper 1 October/November 2006 2 hours Additional Materials: Answer Paper Electronic calculator Graph paper (1 sheet) Mathematical tables READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. At the end of the examination, fasten all your work securely together. This document consists of 6 printed pages and 2 blank pages. SP (SLM) T26709/1 © UCLES 2006 [Turn over www.XtremePapers.com

2 0606/01/O/N/06 © UCLES 2006 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x b b ac a = − − 2 4 2 . Binomial Theorem (a + b)n = an + ( n 1)an–1 b + ( n 2)an–2 b2 + … + ( n r)an–r br + … + bn, where n is a positive integer and ( n r) = n! (n – r)!r!. 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. Formulae for ∆ABC a sin A = b sin B = c sin C . a2 = b2 + c2 – 2bc cos A. ∆ = 1 2 bc sin A.

3 0606/01/O/N/06 © UCLES 2006 [Turn over 1 Express each of the following statements in appropriate set notation. (i) x is not an element of set A. (ii) The number of elements not in set B is 16. (iii) Sets C and D have no common element. [3] 2 –1 –3 –2 0 1 y 60° 120° 90° 30° x The diagram shows part of the graph of y = asin(bx) + c. State the value of (i) a, (ii) b, (iii) c. [3] 3 The equation of a curve is y = 8 (3x – 4)2 . (i) Find the gradient of the curve where x = 2. [3] (ii) Find the approximate change in y when x increases from 2 to 2 + p, where p is small. [2] 4 The vector → OP has a magnitude of 10 units and is parallel to the vector 3i − 4j. The vector → OQ has a magnitude of 15 units and is parallel to the vector 4i + 3j. (i) Express → OP and → OQ in terms of i and j. [3] (ii) Given that the magnitude of → PQ is λ 13 , find the value of λ . [3]

4 0606/01/O/N/06 © UCLES 2006 5 A large airline has a fleet of aircraft consisting of 5 aircraft of type A, 8 of type B, 4 of type C and 10 of type D. The aircraft have 3 classes of seat known as Economy, Business and First. The table below shows the number of these seats in each of the 4 types of aircraft. Class of seat Type of aircraft Economy Business First A 300 60 40 B 150 50 20 C 120 40 0 D 100 0 0 (i) Write down two matrices whose product shows the total number of seats in each class. (ii) Evaluate this product of matrices. On a particular day, each aircraft made one flight. 5% of the Economy seats were empty, 10% of the Business seats were empty and 20% of the First seats were empty. (iii) Write down a matrix whose product with the matrix found in part (ii) will give the total number of empty seats on that day. (iv) Evaluate this total. [6] 6 Given that the coefficient of x2 in the expansion of (k + x)(2 – x 2) 6 is 84, find the value of the constant k. [6] 7 The function f is defined for the domain –3
x
3 by f(x) = 9(x – 1 3) 2 – 11. (i) Find the range of f. [3] (ii) State the coordinates and nature of the turning point of (a) the curve y = f(x), (b) the curve y =
f(x)
. [4]

5 0606/01/O/N/06 © UCLES 2006 [Turn over 8 (a) Solve the equation lg(x + 12) = 1 + lg(2 – x). [3] (b) Given that log2 p = a, log8 q = b and p q = 2c, express c in terms of a and b. [4] 9 A curve has the equation y = 2x – 4 x + 3 . (i) Obtain an expression for dy dx and hence explain why the curve has no turning points. [3] The curve intersects the x-axis at the point P. The tangent to the curve at P meets the y-axis at the point Q. (ii) Find the area of the triangle POQ, where O is the origin. [5] 10 The cubic polynomial f(x) is such that the coefficient of x3 is 1 and the roots of f(x) = 0 are 1, k and k2. It is given that f(x) has a remainder of 7 when divided by x – 2. (i) Show that k3 – 2k2 – 2k – 3 = 0. [3] (ii) Hence find a value for k and show that there are no other real values of k which satisfy this equation. [5] 11 (a) Solve, for 0°
x
360°, the equation 2cot x = 1 + tan x. [5] (b) Given that y is measured in radians, find the two smallest positive values of y such that 6sin(2y + 1) + 5 = 0. [5]

6 0606/01/O/N/06 © UCLES 2006 12 Answer only one of the following two alternatives. EITHER y x B A O C y = 4 – e–2 x The diagram shows part of the curve y = 4 – e–2x which crosses the axes at A and at B. (i) Find the coordinates of A and of B. [2] The normal to the curve at B meets the x-axis at C. (ii) Find the coordinates of C. [4] (iii) Show that the area of the shaded region is approximately 10.3 square units. [5] OR The variables x and y are related by the equation y = 10–Abx, where A and b are constants. The table below shows values of x and y. x 15 20 25 30 35 40 y 0.15 0.38 0.95 2.32 5.90 14.80 (i) Draw a straight line graph of lg y against x, using a scale of 2 cm to represent 5 units on the x-axis and 2 cm to represent 0.5 units on the lg y - axis. [2] (ii) Use your graph to estimate the value of A and of b. [4] (iii) Estimate the value of x when y = 10. [2] (iv) On the same diagram, draw the line representing y5 = 10–x and hence find the value of x for which 10A – x 5 = bx. [3]

8 0606/01/O/N/06 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2006 question paper 0606 ADDITIONAL MATHEMATICS 0606/01 Paper 1, maximum raw mark 80 This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the report on the examination. The grade thresholds for various grades are published in the report on the examination for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 2006 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Mark for a correct result or statement independent of method marks. • When a part of a question has two or more "method" steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. • The symbol √ implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect working. • Note: B2 or A2 means that the candidate can earn 2 or 0. B2/1/0 means that the candidate can earn anything from 0 to 2.

The following abbreviations may be used in a mark scheme or used on the scripts: AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) CAO Correct Answer Only (emphasising that no "follow through" from a previous error is allowed) CWO Correct Working Only – often written by a ‘fortuitous' answer ISW Ignore Subsequent Working MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) Penalties MR –1 A penalty of MR –1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become "follow through √" marks. MR is not applied when the candidate misreads his own figures – this is regarded as an error in accuracy. OW –1,2 This is deducted from A or B marks when essential working is omitted. PA –1 This is deducted from A or B marks in the case of premature approximation. S –1 Occasionally used for persistent slackness – usually discussed at a meeting. EX –1 Applied to A or B marks when extra solutions are offered to a particular equation. Again, this is usually discussed at the meeting.

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